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Quantum geometry and the Schwarzschild singularity

Abhay Ashtekar, Martin Bojowald

TL;DR

This work extends the loop quantum gravity program to the Schwarzschild interior by implementing Kantowski-Sachs symmetry and quantizing the reduced theory. It shows that a self-adjoint quantum Hamiltonian constraint can be defined using holonomies and the quantum area gap, producing a discrete evolution that traverses the classical singularity rather than terminating there. The kinematical quantum geometry regularizes inverse-triad quantities and suggests a quantum bounce between two classical regions, offering a potential resolution to information-loss concerns in black hole evaporation. While the results are promising, the model remains highly simplified, and substantial future work is required to incorporate regions outside the horizon and additional degrees of freedom to assess robustness and physical viability.

Abstract

In homogeneous cosmologies, quantum geometry effects lead to a resolution of the classical singularity without having to invoke special boundary conditions at the singularity or introduce ad-hoc elements such as unphysical matter. The same effects are shown to lead to a resolution of the Schwarzschild singularity. The resulting quantum extension of space-time is likely to have significant implications to the black hole evaporation process. Similarities and differences with the situation in quantum geometrodynamics are pointed out.

Quantum geometry and the Schwarzschild singularity

TL;DR

This work extends the loop quantum gravity program to the Schwarzschild interior by implementing Kantowski-Sachs symmetry and quantizing the reduced theory. It shows that a self-adjoint quantum Hamiltonian constraint can be defined using holonomies and the quantum area gap, producing a discrete evolution that traverses the classical singularity rather than terminating there. The kinematical quantum geometry regularizes inverse-triad quantities and suggests a quantum bounce between two classical regions, offering a potential resolution to information-loss concerns in black hole evaporation. While the results are promising, the model remains highly simplified, and substantial future work is required to incorporate regions outside the horizon and additional degrees of freedom to assess robustness and physical viability.

Abstract

In homogeneous cosmologies, quantum geometry effects lead to a resolution of the classical singularity without having to invoke special boundary conditions at the singularity or introduce ad-hoc elements such as unphysical matter. The same effects are shown to lead to a resolution of the Schwarzschild singularity. The resulting quantum extension of space-time is likely to have significant implications to the black hole evaporation process. Similarities and differences with the situation in quantum geometrodynamics are pointed out.

Paper Structure

This paper contains 12 sections, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical Theory
  3. Symmetry reduction
  4. The reduced phase space
  5. Quantum Kinematics
  6. Basics
  7. Quantum geometry
  8. Quantum Dynamics
  9. Strategy
  10. Quantum Hamiltonian Constraint
  11. Absence of Singularity
  12. Discussion

Figures (1)

  • Figure 1: Dynamical trajectories in the $p_b-p_c$ plane. Each trajectory reaches the maximum value $m$ of $p_b$ and meets the $p_c =0$ axis only at the point where $p_b$ also vanishes (i.e., at the 'origin'). Solutions related by the reflection-symmetry $p_c \rightarrow -p_c$ define the same metric but carry triads of opposite orientation. For simplicity the '$b$-parity gauge' is fixed by requiring $p_b \ge 0$.